3.5 The Partition Coefficient

In our study of equilibria, we have looked at systems involving gases, aqueous solutions, and solid precipitates. Another important type of equilibrium occurs when a solute is distributed between two immiscible (non-mixing) liquids. This concept is crucial in many fields, especially in the pharmaceutical industry for drug design and in analytical chemistry for separation techniques. The quantitative measure of this distribution is called the partition coefficient.

Definition

The partition coefficient (K_pc) is defined as the ratio of the concentrations of a solute in two different immiscible solvents in contact with each other when equilibrium has been established at a particular temperature.

Simple Explanation: If you have a solute dissolved in a mixture of two liquids that don't mix (like oil and water), the solute will distribute itself between the two layers. The partition coefficient is a number that tells you how much of the solute prefers one liquid over the other.

Core Concept

The core concept is dynamic equilibrium between two immiscible phases. When a solute is added to a system of two immiscible solvents, it dissolves in both. Initially, the concentration in one solvent might be higher, so the solute molecules move from the higher concentration phase to the lower concentration phase. Eventually, a dynamic equilibrium is reached where the rate of molecules moving in one direction equals the rate of molecules moving in the opposite direction. At this point, the ratio of the concentrations in the two solvents is constant at a given temperature. This constant is the partition coefficient.

Detailed Explanation

The Distribution Law

If you put two immiscible liquids, like ether and water, in a separatory funnel and shake, they will form two layers on standing. Ether is less dense, so it forms the upper layer.

Assume that a solute X dissolves in both liquids. If you prepare a solution of known concentration of X in water and mix it with ether in a separatory funnel and shake well, on standing, the two layers separate again. If you now determine the concentration of the solute in the water layer, you will notice a decrease in concentration. This is because the solute is distributed between the two solvents. A dynamic equilibrium is created between these two solutions. At equilibrium, the concentration of X becomes constant in both layers.

X(in water) ⇌ X(in ether)

The equilibrium constant for this process is called the partition coefficient and is represented by the symbol K_pc:

K_pc = [X in ether][X in water]

This equilibrium constant has a constant value at a constant temperature. The partition coefficient is a ratio of two concentrations, so it has no units.

The Nernst Distribution Law

The distribution law, or Nernst distribution law, gives a generalization that governs the distribution of a solute between two immiscible solvents. This law was first given by Nernst, who studied the partitioning of several solutes between various suitable pairs of solvents.

Calculation of Partition Coefficient (Activity 3.1)

The textbook provides a detailed activity to calculate the partition coefficient of methylamine between an organic solvent and water.

• Volume of aqueous methylamine solution: 100 cm³
• Initial concentration of methylamine: 0.150 mol dm^-3
• Volume of organic solvent: 75.0 cm³
• Volume of aqueous layer titrated: 25.0 cm³
• Volume of HCl used for titration: 7.05 cm³
• Concentration of HCl: 0.225 mol dm^-3

Step-by-Step Solution:

Step 1: Write the equilibrium equation. CH_3NH_{2(aq)} \rightleftharpoons CH_3NH_2(\text{organic solvent})

Step 2: Write the K_pc expression. K_pc = [CH_3NH_2(organic layer)][CH_3NH_2(aq)]

Step 3: Determine the total moles of methylamine in the original solution. Total moles of CH_3NH_2 = 0.150 mol dm^-3 × 0.100 dm³ = 0.015 mol

Step 4: Determine the moles of methylamine in the aqueous layer after equilibrium. - The aqueous layer was titrated with HCl: CH_3NH_2(aq) + HCl(aq) → CH_3NH_3Cl(aq) - Moles of HCl used = 0.225 mol dm^-3 × 0.00705 dm³ = 0.001586 mol - This is the moles of methylamine in 25.0 cm³ of the aqueous layer. - Moles of methylamine in 100 cm³ of the aqueous layer: Moles = 0.001586 × 10025 = 0.00634 mol

Step 5: Determine the moles of methylamine in the organic layer. Moles in organic layer = Total moles - moles in aqueous layer = 0.015 - 0.00634 = 0.00866 mol

Step 6: Calculate the concentrations in each layer. Concentration in aqueous layer = 0.00634 mol0.100 dm³ = 0.0634 mol dm^-3 Concentration in organic layer = 0.00866 mol0.075 dm³ = 0.115 mol dm^-3 (Note: The textbook's calculation gives 0.116 mol dm^-3 for the organic layer and 0.063 mol dm^-3 for the aqueous layer. The value of 0.0364 in the textbook's final step is a typographical error. The correct calculation uses the concentrations derived above.)

Step 7: Calculate the partition coefficient (K_pc). K_pc = 0.1160.063 ≈ 1.84

Conclusion: Since the value of K_pc is larger than 1, methylamine is more soluble in the organic solvent than in water.

Intermediate Level

The partition coefficient (K_pc) is an equilibrium constant for the distribution of a solute between two immiscible solvents. It is defined as the ratio of the concentration of the solute in one solvent to its concentration in the other solvent at equilibrium. It has no units because it is a ratio of concentrations. The value of K_pc depends on the temperature and the nature of the solute and the two solvents. If K_pc > 1, the solute is more soluble in the solvent in the numerator. If K_pc < 1, it is more soluble in the solvent in the denominator. This concept is widely used in the extraction of compounds from one solvent to another, and in understanding how drugs are distributed in different parts of the body (e.g., crossing the lipid bilayer of cell membranes).

Important Points

• - Partition Coefficient (K_pc): Ratio of solute concentrations in two immiscible solvents at equilibrium.
• - Distribution Law (Nernst): Describes the distribution of a solute between two immiscible solvents.
• - No Units: It is a ratio of concentrations.
• - Temperature dependent: K_pc has a constant value at a given temperature.
• - Interpretation: K_pc > 1 means the solute is more soluble in the numerator solvent. K_pc < 1 means it is more soluble in the denominator solvent.
• - Applications: Drug design (log P), extraction techniques.
• - Calculation: Requires knowing the total moles, the moles in one layer (often determined by titration), and the volumes of both layers.

Remember

• - K_pc = [Solute in solvent 1][Solute in solvent 2] at equilibrium.
• - No units.
• - K_pc > 1: Solute prefers solvent 1 (numerator).
• - K_pc < 1: Solute prefers solvent 2 (denominator).
• - K_pc depends on temperature.
• - Used in drug design and extraction.
• - Nernst distribution law: The basis for the partition coefficient.

Common Mistakes

• - Forgetting that the concentration must be at equilibrium. The calculation is based on the concentrations of the solute in both layers after they have been mixed and allowed to separate.
• - Using the wrong volumes. Make sure you use the total volume of each layer, not the volume of the aliquot that you titrated. In the example, the aqueous layer volume is 100 cm³, even though only 25 cm³ was titrated.
• - Ignoring the units. The partition coefficient is a ratio of concentrations, so it is unitless. However, it is crucial to use the same units for both concentrations (e.g., mol/dm³) when calculating the ratio.
• - Assuming that the solute does not dissociate. If the solute is a weak acid or base, it may dissociate in water, which would affect the calculation. In such cases, you need to consider the dissociation equilibrium.
• - Confusing the partition coefficient with solubility. The partition coefficient is a ratio of concentrations, not the absolute solubility of a solute. A solute can have a high partition coefficient even if its overall solubility is low.

Quick Revision

The partition coefficient (K_pc) is an equilibrium constant that describes the distribution of a solute between two immiscible solvents. It is defined as the ratio of the solute's concentrations in the two solvents at equilibrium. It has no units and is constant at a given temperature. A value greater than 1 indicates that the solute is more soluble in the solvent in the numerator, while a value less than 1 indicates it is more soluble in the denominator. The partition coefficient is used in drug design and extraction techniques. The textbook provides an example of calculating the partition coefficient of methylamine between an organic solvent and water using titration data.

60-Second Revision

The partition coefficient (K_pc) is the ratio of the concentrations of a solute in two immiscible solvents at equilibrium. It is unitless and constant at a given temperature. If K_pc > 1, the solute prefers the solvent in the numerator; if K_pc < 1, it prefers the denominator. It is used in drug design and extraction. The Nernst distribution law governs this equilibrium.

Frequently Asked Questions (FAQs)

Q1. What is the partition coefficient (K_pc)? Answer: The partition coefficient (K_pc) is the equilibrium constant for the distribution of a solute between two immiscible solvents. It is defined as the ratio of the concentration of the solute in one solvent to its concentration in the other solvent at equilibrium, at a given temperature.

Q2. Why does the partition coefficient have no units? Answer: The partition coefficient is a ratio of two concentrations (K_pc = [solute]₁ / [solute]₂). Since the units of concentration in the numerator and denominator are the same (e.g., mol/dm³), they cancel out, leaving a unitless number.

Q3. What does a partition coefficient greater than 1 indicate? Answer: A partition coefficient greater than 1 indicates that the solute is more soluble in the solvent in the numerator. For example, if K_pc = [solute in ether] / [solute in water] > 1, it means the solute is more soluble in ether than in water.

Q4. What is the Nernst distribution law? Answer: The Nernst distribution law (or partition law) states that a solute distributes itself between two immiscible solvents in such a way that the ratio of its concentrations in the two solvents is a constant at a given temperature, provided the solute is in the same molecular form in both solvents.

Q5. How is the partition coefficient calculated? Answer: The partition coefficient is calculated by determining the concentration of the solute in each layer after equilibrium is reached. This often involves titrating one of the layers to find the concentration of the solute. The concentrations are then substituted into the K_pc expression.

Q6. Explain how the partition coefficient can be used to estimate the distribution of drugs within the body. Answer: In the body, drugs must be distributed between aqueous (blood plasma) and lipid (cell membranes) phases. A partition coefficient can predict how a drug will distribute between these phases. If a drug has a very high partition coefficient for lipids, it may accumulate in fatty tissues and have a longer duration of action. If it is too hydrophilic (low K_pc for lipids), it may not cross cell membranes effectively. Thus, K_pc is a key parameter in drug design for optimizing absorption and distribution.

• Temperature: The equilibrium constant is temperature-dependent.
• Nature of the solute: Its polarity and ability to interact with the solvents.
• Nature of the solvents: Their polarity and ability to dissolve the solute.
• Presence of other solutes: These can affect the activity of the solute.

Q8. What are the limitations of the Nernst distribution law? Answer: The Nernst distribution law assumes that the solute exists in the same molecular form in both solvents. If the solute dissociates or associates in one of the solvents, the simple distribution law must be modified to account for these equilibria. Additionally, the law is most accurate for dilute solutions where activity coefficients are close to 1.

Q9. How does the partition coefficient relate to the polarity of the solute and the solvents? Answer: The partition coefficient indicates the relative affinity of a solute for two solvents. A polar solute will have a higher partition coefficient in favor of a polar solvent (e.g., water). A non-polar solute will have a higher partition coefficient in favor of a non-polar solvent (e.g., ether or octanol). This is a key principle in choosing the right solvent for extraction.

Multiple Choice Questions (MCQs)

Q1. What is the partition coefficient defined as? A. The ratio of solute concentration in one solvent to the other. B. The ratio of solute concentration in a single solvent. C. The ratio of solute mass in one solvent to the other. D. The ratio of solute mass in a single solvent. Correct Answer: A Explanation: The partition coefficient is defined as the ratio of the concentrations of a solute in two different immiscible solvents in contact with each other when equilibrium has been established.

• Q2. What is the unit of the partition coefficient?
• A. mol/dm³
• B. g/L
• C. It has no units.
• D. mol/s
• Correct Answer: C
• Explanation: The partition coefficient is a ratio of two concentrations, so the units cancel out, leaving a unitless number.

Q3. Which type of solvent would favor the partitioning of a polar solute? A. Non-polar solvent B. Hydrophobic solvent C. Polar solvent D. Aprotic solvent Correct Answer: C Explanation: A polar solute has a greater affinity for a polar solvent due to dipole-dipole interactions or hydrogen bonding, resulting in a higher partition coefficient in favor of the polar solvent.

Q4. If a solute has a partition coefficient of 10 in favor of an organic solvent, where is it more soluble? A. In the aqueous solvent B. In the organic solvent C. Equally soluble in both D. It is insoluble Correct Answer: B Explanation: A partition coefficient > 1 indicates that the solute is more soluble in the solvent in the numerator. For K_pc = [organic]/[aqueous] = 10, the solute is 10 times more concentrated in the organic layer.

Q5. What is the Nernst distribution law? A. A law that states that the ratio of solute concentrations in two immiscible solvents is constant at a given temperature. B. A law that states that the rate of diffusion is proportional to the concentration gradient. C. A law that states that the pressure of a gas is inversely proportional to its volume. D. A law that states that the enthalpy change of a reaction is independent of the path. Correct Answer: A Explanation: The Nernst distribution law states that at a constant temperature, a solute distributes itself between two immiscible solvents such that the ratio of its concentrations in the two solvents is a constant.

Q6. In the context of drug design, what does the partition coefficient help predict? A. The color of the drug. B. The solubility of the drug in water and lipids. C. The melting point of the drug. D. The smell of the drug. Correct Answer: B Explanation: The partition coefficient is used to predict how a drug will distribute between aqueous (blood) and lipid (cell membrane) phases, which is crucial for understanding its absorption and distribution in the body.

Q7. If the partition coefficient of a solute between water and octanol is 0.1 (with water in the numerator), where is it more soluble? A. In water B. In octanol C. Equally soluble in both D. It is insoluble Correct Answer: B Explanation: K_pc = [water]/[octanol] = 0.1. This means the solute is 10 times more concentrated in octanol than in water. So it is more soluble in octanol.

Q8. What does a partition coefficient of 1 indicate? A. The solute is equally soluble in both solvents. B. The solute is more soluble in the first solvent. C. The solute is more soluble in the second solvent. D. The solute is insoluble in both solvents. Correct Answer: A Explanation: A partition coefficient of 1 means the concentration of the solute in both solvents is equal at equilibrium, indicating that it is equally soluble in both.

Q9. Why is the partition coefficient important in the pharmaceutical industry? A. It is used to determine the color of drugs. B. It is used to determine the dosage of drugs. C. It is used to predict the absorption and distribution of drugs in the body. D. It is used to determine the melting point of drugs. Correct Answer: C Explanation: The partition coefficient is a key parameter in drug design for predicting how a drug will distribute between aqueous and lipid phases, affecting its absorption, distribution, metabolism, and excretion.

Q10. What is the formula for the partition coefficient of a solute X between water and ether? A. K_pc = [X in ether]/[X in water] B. K_pc = [X in water]/[X in ether] C. K_pc = [X in water] + [X in ether] D. K_pc = [X in water] × [X in ether] Correct Answer: A Explanation: The partition coefficient is the ratio of the concentration of the solute in the two solvents. The choice of which concentration is in the numerator depends on how the coefficient is defined, but it is typically defined as the ratio of the concentration in the organic solvent to that in the aqueous solvent. In the textbook, it is defined as [X in ether]/[X in water].

Multiple Choice Questions (MCQs)

• Q1. What is the partition coefficient (K_pc) a measure of?
• A. The solubility of a solute in a single solvent.
• B. The distribution of a solute between two immiscible solvents.
• C. The rate of dissolution of a solute.
• D. The temperature of a solution.
• Correct Answer: B
• Explanation: The partition coefficient is a measure of the distribution of a solute between two immiscible solvents at equilibrium.

• Q2. What is the unit of the partition coefficient?
• A. mol/dm³
• B. g/L
• C. Unitless
• D. mol/s
• Correct Answer: C
• Explanation: The partition coefficient is a ratio of two concentrations, so its units cancel out, making it a unitless quantity.

Q3. If a solute has a partition coefficient of 0.5 in favor of an organic solvent (defined as [organic]/[water]), where is it more soluble? A. In the organic solvent B. In water C. Equally soluble in both D. It is insoluble Correct Answer: B Explanation: K_pc = [organic]/[water] = 0.5. This means the concentration in the organic solvent is half of that in water. Therefore, the solute is more soluble in water.

Q4. What is the Nernst distribution law? A. A law that states that the ratio of solute concentrations in two immiscible solvents is constant at a given temperature. B. A law that states that the rate of diffusion is proportional to the concentration gradient. C. A law that states that the pressure of a gas is inversely proportional to its volume. D. A law that states that the enthalpy change of a reaction is independent of the path. Correct Answer: A Explanation: The Nernst distribution law states that at a constant temperature, a solute distributes itself between two immiscible solvents such that the ratio of its concentrations in the two solvents is a constant.

Q5. In the context of drug design, what does a high partition coefficient (in favor of lipids) indicate? A. The drug is highly water-soluble. B. The drug is highly lipid-soluble. C. The drug is insoluble in both. D. The drug will not cross cell membranes. Correct Answer: B Explanation: A high partition coefficient in favor of lipids means the drug is more soluble in lipid (fat) phases, such as cell membranes. This is important for its ability to cross membranes and reach its target site.

Q6. What is the partition coefficient formula for a solute X between water and ether? A. K_pc = [X in ether]/[X in water] B. K_pc = [X in water]/[X in ether] C. K_pc = [X in water] + [X in ether] D. K_pc = [X in water] × [X in ether] Correct Answer: A Explanation: The partition coefficient is usually defined as the ratio of the concentration of the solute in the organic solvent to its concentration in the aqueous solvent. The textbook defines it as [X in ether]/[X in water].

Q7. Why is the partition coefficient important in the pharmaceutical industry? A. It is used to determine the color of drugs. B. It is used to predict the absorption and distribution of drugs in the body. C. It is used to determine the melting point of drugs. D. It is used to determine the dosage of drugs. Correct Answer: B Explanation: The partition coefficient is a key parameter in drug design for predicting how a drug will distribute between aqueous and lipid phases, affecting its absorption, distribution, metabolism, and excretion.

Q8. What does a partition coefficient of 1 indicate? A. The solute is equally soluble in both solvents. B. The solute is more soluble in the first solvent. C. The solute is more soluble in the second solvent. D. The solute is insoluble in both solvents. Correct Answer: A Explanation: A partition coefficient of 1 means the concentration of the solute in both solvents is equal at equilibrium, indicating that it is equally soluble in both.

Q9. In the calculation of the partition coefficient of methylamine, what was the total number of moles of methylamine in the original solution? A. 0.015 mol B. 0.00634 mol C. 0.00866 mol D. 0.001586 mol Correct Answer: A Explanation: The initial concentration was 0.150 mol dm^-3 and the volume was 100 cm³ (0.100 dm³), so the total moles = 0.150 × 0.100 = 0.015 mol.

Q10. In the same calculation, what was the final concentration of methylamine in the organic layer? A. 0.063 mol dm^-3 B. 0.116 mol dm^-3 C. 0.150 mol dm^-3 D. 0.225 mol dm^-3 Correct Answer: B Explanation: The moles of methylamine in the organic layer were 0.00866 mol, and the volume was 75.0 cm³ (0.075 dm³). So the concentration is 0.008660.075 = 0.115 mol dm^-3, approximately 0.116 mol dm^-3.